# Complex Arithmetic/Examples/(5 + 5i) (3 - 4i)^-1 + 20 (4 + 3i)^-1

## Example of Complex Arithmetic

$\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i} = 3 - i$

## Proof

 $\ds \dfrac {5 + 5 i} {3 - 4 i}$ $=$ $\ds \dfrac {\paren {5 + 5 i} \paren {3 + 4 i} } {\paren {3 - 4 i} \paren {3 + 4 i} }$ multiplying top and bottom by $3 + 4 i$ $\ds$ $=$ $\ds \dfrac {15 + 15 i + 20 i + 20 i^2} {3^2 + 4^2}$ Definition of Complex Multiplication $\ds$ $=$ $\ds \dfrac {-5 + 35 i} {25}$ simplifying

Then:

 $\ds \dfrac {20} {4 + 3 i}$ $=$ $\ds \dfrac {20 \paren {4 - 3 i} } {\paren {4 + 3 i} \paren {4 - 3 i} }$ multiplying top and bottom by $4 - 3 i$ $\ds$ $=$ $\ds \dfrac {80 - 60 i} {4^2 + 3^2}$ Definition of Complex Multiplication $\ds$ $=$ $\ds \dfrac {80 - 60 i} {25}$ simplifying

Finally:

 $\ds \dfrac {-5 + 35 i} {25} + \dfrac {80 - 60 i} {25}$ $=$ $\ds \dfrac {\paren {-5 + 80} + \paren {35 - 60} i} {25}$ Definition of Complex Addition $\ds$ $=$ $\ds \dfrac {75 - 25 i} {25}$ $\ds$ $=$ $\ds 3 - i$ simplifying

$\blacksquare$